The population of a slowly growing bacterial colony after t hours is given by p(t)=3t2+29t+150 . Find the growth rate after 4 hours?
I assume you meant
p(t)=3t^2+29t+150
then p'(t) = 6t + 29 and
p'(4) = 6(4)+29
To find the growth rate after 4 hours, we need to find the derivative of the population function with respect to time.
The population function is given by p(t) = 3t^2 + 29t + 150.
Taking the derivative of p(t) with respect to t will give us the rate of change of the population, which represents the growth rate.
p'(t) = d/dt (3t^2 + 29t + 150)
To find the derivative, we can apply the power rule and the sum rule of derivatives.
The power rule states that for a function f(t) = c*t^n, where c is a constant and n is a real number, the derivative is given by f'(t) = c*n*t^(n-1).
Applying the power rule to each term of the population function, we have:
p'(t) = d/dt (3t^2) + d/dt (29t) + d/dt (150)
= 6t + 29 + 0 (since the derivative of a constant is 0)
= 6t + 29
Now, to find the growth rate after 4 hours, we substitute t = 4 into the derived equation.
p'(4) = 6(4) + 29
= 24 + 29
= 53
Therefore, the growth rate after 4 hours is 53 bacteria per hour.